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% Document Info
\author{Zulle}
\pdfinfo{
  /Title (geometria-analitica-e-algebra-linear-vetores.pdf)
  /Creator (Cheatography)
  /Author (Zulle)
  /Subject (Geometria Analítica e Álgebra Linear - Vetores Cheat Sheet)
}

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\fancyhead[L]{
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Geometria Analítica e Álgebra Linear - Vetores Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Zulle} via \textcolor{DarkBackground}{\uline{cheatography.com/126577/cs/24557/}}}
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\end{multicols}}

\fancyfoot[L]{ \footnotesize
\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Zulle \\
  \uline{cheatography.com/zulle} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 30th September, 2020.\\
   Updated 30th September, 2020.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
\vfill
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
\end{tabulary}
\end{multicols}}




\begin{document}
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\raggedcolumns

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% from this page to resize cheat sheet text:
% www.emerson.emory.edu/services/latex/latex_169.html
\footnotesize % Small font.

\begin{multicols*}{2}

\begin{tabularx}{8.4cm}{x{4 cm} x{4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Componentes do Vetor}}  \tn
% Row 0
\SetRowColor{LightBackground}
A = (x`a`,y`a`) & B = (x`b`,y`b`) \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{\{\{ac\}\} \textasciitilde{}\textasciitilde{}AB\textasciitilde{}\textasciitilde{} = B - A}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Norma}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{"módulo da distância entre os pontos do vetor"} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{\{\{ac\}\} ||\textasciitilde{}\textasciitilde{}w\textasciitilde{}\textasciitilde{}|| = ( (x`2`-x`1`)\textasciicircum{}2\textasciicircum{}+(y`2`-y`1`)\textasciicircum{}2\textasciicircum{}+(z`2`-z`1`)\textasciicircum{}2\textasciicircum{} )\textasciicircum{}1/2\textasciicircum{}} \tn 
% Row Count 3 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{1.84 cm} x{6.16 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Operações}}  \tn
% Row 0
\SetRowColor{LightBackground}
soma & \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}+\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = (v`x`+w`x`, v`y`+w`y`, v`z`+w`z`) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\{\{ar\}\} obs.: & (\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}+\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) + \textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{} = \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} + (\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} + \textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{}) \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\seqsplit{diferença} & \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}+\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = (v`x`-w`x`, v`y`-w`y`, v`z`-w`z`) \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\{\{ar\}\} obs.: & (\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} - \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) = \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} + (-\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{2.32 cm} x{5.68 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Vetores}}  \tn
% Row 0
\SetRowColor{LightBackground}
simétrico & \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} + (- \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}) = 0 \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
unitário & \textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{} = ( \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} / ||\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}|| ) \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\{\{ar\}\} obs.: & ||\textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{}|| = 1 \tn 
% Row Count 5 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{2.8 cm} x{5.2 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Multiplicação por Escalar}}  \tn
% Row 0
\SetRowColor{LightBackground}
\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} = a\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} & -\textgreater{} são vetores colineares \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{a(b\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}) = ab(\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{})} \tn 
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% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{a(\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} + \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) = a\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} +a\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{( a+b ) \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} = a\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} + b\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}} \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{4.56 cm} x{3.44 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Produto Escalar}}  \tn
% Row 0
\SetRowColor{LightBackground}
\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} . \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = 0 & se \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} ou \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = 0 \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} . \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = || \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} ||.|| \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} ||.cos(t)} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
t\textasciicircum{}o\textasciicircum{} = arccos (\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} . \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) & \{\{ar\}\} achar o ângulo \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{. . . . || \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} ||.|| \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} ||} \tn 
% Row Count 6 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{1. achar derivada de g(x)} \tn 
% Row Count 7 (+ 1)
% Row 5
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} . \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} . \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}} \tn 
% Row Count 8 (+ 1)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{\textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{} (\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} + \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) = \textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{}. \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} + \textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{}. \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}} \tn 
% Row Count 10 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{5.6 cm} x{2.4 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{Produto Vetorial}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{\{\{ac\}\} || \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} || = ||\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{}|| . ||\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}|| . sen(t)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{8.4cm}}{\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = det ( \textasciitilde{}\textasciitilde{}i\textasciitilde{}\textasciitilde{} , \textasciitilde{}\textasciitilde{}j\textasciitilde{}\textasciitilde{} , \textasciitilde{}\textasciitilde{}k\textasciitilde{}\textasciitilde{} )} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = -(\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{})} \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
|| \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} || = ( i\textasciicircum{}2\textasciicircum{}+j\textasciicircum{}2\textasciicircum{}+k\textasciicircum{}2\textasciicircum{})\textasciicircum{}1/2\textasciicircum{} & \textless{}- área vetorial \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Produto Misto}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{\{\{ac\}\} (\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) . \textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{} = det ( U`x` U`y` U`z`)} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{\{\{ac\}\}\_ \_ \_ \_ \_ \_ \_ \_ \_ \_  ( V`x` V`y` V`z`)} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{\{\{ac\}\}\_ \_ \_ \_ \_ \_ \_ \_ \_ \_  \_( W`x` W`y` W`z`)} \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{obs.: o modulo do produto misto é igual ao {\bf{volume}} do paralelepípedo formado por tais vetores.}  \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{Projeção Ortogonal}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{"parte de \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} que está sobre \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}" -\textgreater{} \textasciitilde{}\textasciitilde{}V`x`\textasciitilde{}\textasciitilde{} = a\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{8.4cm}}{proj` \textasciitilde{}\textasciitilde{}w\textasciitilde{}\textasciitilde{} `\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} = \textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} . \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} . \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{8.4cm}}{\_ \_ \_ \_ \_ \_  || \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}\textasciicircum{}2\textasciicircum{} ||} \tn 
% Row Count 4 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{8.4cm}}{\bf\textcolor{white}{NOTAS}}  \tn
\SetRowColor{LightBackground}
\mymulticolumn{1}{p{8.4cm}}{\vspace{1px}\centerline{\includegraphics[width=5.1cm]{/web/www.cheatography.com/public/uploads/zulle_1601468118_vetores.jpg}}} \tn 
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{x{3.52 cm} x{4.48 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{NOTAS}}  \tn
% Row 0
\SetRowColor{LightBackground}
\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = 0 & se forem paralelos (\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} = a\textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} . \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{} = 0 & se forem perpendiculares \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\textasciitilde{}\textasciitilde{}U\textasciitilde{}\textasciitilde{} . (\textasciitilde{}\textasciitilde{}V\textasciitilde{}\textasciitilde{} x \textasciitilde{}\textasciitilde{}W\textasciitilde{}\textasciitilde{}) = 0 & se forem coplanares \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{8.4cm}{p{0.8 cm} p{0.8 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{8.4cm}}{\bf\textcolor{white}{NOTAS}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{8.4cm}}{||V||} \tn 
% Row Count 1 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}